Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems

TL;DR

This paper introduces a coordinate-free method for deriving higher-order adiabatic invariants in generalized slow-fast Hamiltonian systems, with explicit formulas and applications to physical models like the elastic pendulum and charged particles.

Contribution

It develops a global averaging approach to construct approximate first integrals without coordinate dependence, including explicit second-order formulas for complex systems.

Findings

Derived explicit formulas for second-order adiabatic invariants.

Applied the method to elastic pendulum and charged particle models.

Demonstrated the effectiveness of the approach in physical systems.

Abstract

We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with $\mathbb{S}^1 -$symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.